Limit curves for zeros of sections of exponential integrals

We are interested in studying the asymptotic behavior of the zeros of partial sums of power series for a family of entire functions defined by exponential integrals. The zeros grow on the order of O(n), and after rescaling we explicitly calculate their limit curve. We find that the rate that the zeros approach the curve depends on the order of the singularities/zeros of the integrand in the exponential integrals. As an application of our findings we derive results concerning the zeros of partial sums of power series for Bessel functions of the first kind.Comment: 19 pages, 5 figures. arXiv admin note: text overlap with arXiv:1208.518

Bayesian optimization for the inverse scattering problem in quantum reaction dynamics

We propose a machine-learning approach based on Bayesian optimization to build global potential energy surfaces (PES) for reactive molecular systems using feedback from quantum scattering calculations. The method is designed to correct for the uncertainties of quantum chemistry calculations and yield potentials that reproduce accurately the reaction probabilities in a wide range of energies. These surfaces are obtained automatically and do not require manual fitting of the {\it ab initio} energies with analytical functions. The PES are built from a small number of {\it ab initio} points by an iterative process that incrementally samples the most relevant parts of the configuration space. Using the dynamical results of previous authors as targets, we show that such feedback loops produce accurate global PES with 30 {\it ab initio} energies for the three-dimensional H + H$_2$ $\rightarrow$ H$_2$ + H reaction and 290 {\it ab initio} energies for the six-dimensional OH + H$_2$ $\rightarrow$ H$_2$O + H reaction. These surfaces are obtained from 360 scattering calculations for H$_3$ and 600 scattering calculations for OH$_3$. We also introduce a method that quickly converges to an accurate PES without the {\it a priori} knowledge of the dynamical results. By construction, our method illustrates the lowest number of potential energy points (i.e. the minimum information) required for the non-parametric construction of global PES for quantum reactive scattering calculations.Comment: 9 pages, 8 figure